The Broad Hinge
How to fold it
- Print the sheet at 100% and cut cleanly along the outer border of the 6×5 grid
- With a ruler and an empty ballpoint, score every one of the 49 lines before folding, keeping the 24 mountain and 25 valley assignments separate in your mind
- Accordion-fold the first crease family across the grid, orange lines coming toward you, blue lines going away
- Fold the closed strip along the second crease family, alternating direction consistently corner to corner
- Open the sheet flat and correct any crease that fights its printed color, paying attention near the 38° and 24° corners where the fold range tightens
- Open and close the whole pattern a few times through its full 90.0° range so the paper learns the folds.
Pre-creasing every line before you collapse is the whole trick. Paper remembers.
The story below, read word for word by a synthetic voice over this fold's own music.
A door in an old house does not swing so much as decide. You push it, and for the first quarter of its travel nothing happens except a shift in the air; then the weight finds its axis and the whole panel moves as one thing. I spent a week with a door like that in a rented room, opening and closing it while I should have been working, trying to name what I liked about it. It was the broadness of the hinge — the sense that the turning was not concentrated in a small pin but spread across the width of the wood.
The Broad Hinge, catalogued here as FSO-0001, came out of that week. It is our own pattern: designed at Foldsong in 2026, generated and checked by our software, with the measurements printed in the credits below so you can see what was tested and what was not.
Where the Difficulty Sits
Origami tessellations that fold flat are governed by conditions that have been understood for some time. Kawasaki gave the angle condition at a vertex: the alternating sums of the angles around it must balance. Maekawa gave the count: at a flat-foldable interior vertex, the number of mountains and the number of valleys differ by two. These are not aesthetic preferences. They are arithmetic, and a pattern either satisfies them or it does not.
The harder question comes after. A sheet can satisfy the flat-folding conditions and still refuse to move as a mechanism — it may need the paper to stretch, or bend a panel, or pass through an impossible intermediate shape. The classical Miura pattern, described by Miura, is the case where everything cooperates: one family of parallelograms, one clean degree of freedom, a fold that behaves the same everywhere. Most departures from it lose that cooperation quietly.
The Broad Hinge belongs to the generalized translational surface family. It is not a Miura. The task was to leave that comfortable case without losing the motion.
What the Sheet Does
On a 6×5 grid, the pattern carries 42 vertices and 71 creases, of which 20 vertices are interior. The assignment resolves to 24 mountains, 25 valleys, and 22 boundary edges. The corner angles are 135°, 38°, 45° and 142° — the pair of shallow angles is what widens the hinge line and keeps the turning from collecting at a point.
Folding range is 90.0°. We sampled 21 intermediate states along that range and measured the rigid-folding deviation at each; the worst case came in at 1.2e-15, well inside our threshold of 1e-9. Both crease families flex, so the sheet is not merely hinging along one direction while the other stays rigid.
Relief per cell measures 0.62. For reference, a 60° Miura sits at 0.87. The Broad Hinge is the flatter object, and that is the trade: less height for a longer, wider turn. In the hand it feels less like a bellows and more like a panel deciding to move.
Some hinges are a pin; this one is a plank.